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LieAlgebras[TensorProduct] - form the tensor product representation for a list of representations of a Lie algebra; form various tensor product representations from a single representation of a Lie algebra
Calling Sequences
TensorProduct(R, W)
TensorProduct(rho, T, W)
Parameters
R - a list R = [rho1, rho2, ...] of representations rho1, rho2, ... of a Lie algebra g on vector spaces V1, V2, ...
W - a Maple name or string, the name of the frame for the representation space for the tensor product representation
rho - a representation of a Lie algebra g on a vector space V
T - a list of linearly independent type (r,s) tensors on V defining a subspace of tensors invariant under the induced representation of rho
Description
Let W = V1 * V2 *... be the tensor product of the vector spaces V1, V2, ... The dimension of W is the product of the dimensions of the vector spaces V1, V2, ... Then the tensor product of the representations rho1, rho2,... is the representation of Lie algebra rho of g on W defined by rho(x)(y1*y2* ...) = rho(x)(y1)*y2* ... + y1 *rho(x)(y2)*... + ... where y1 in V1, y2 in V2, ... and x in g.
The second calling sequence returns a p dimensional representation of rho, where p is the number of elements in the list T, defined by the restriction to T of the representation of rho on the space T^r_s(V) of type (r,s) tensors on V. For example, T may be a basis for all symmetric or skew-symmetric tensors of a given rank.
Examples
Example 1.
Define the standard representation and the adjoint representation for sl2. Then form the tensor product representation. First, setup the representation spaces.
Define the standard representation.
Define the adjoint representation using the Adjoint command.
We will need a 6 dimensional vector space for the representation space for the tensor product of rho1 and rho2.
Use the Query command to verify that rho1 is a representation.
Example 2.
Compute the representation of rho1 (the standard representation of sl2) on the 3rd symmetric product Sym^3(V1) of V1. Use the GenerateSymmetricTensors command to generate a basis T1 for this tensor space.
We will need a 4 dimensional vector space for the representation space.
Example 3.
Compute the representation of rho1 (the standard representation of sl2) on the 2nd exterior product of the 3rd symmetric product Lambda^2(Sym^3(V1)).
We will need a 6 dimensional vector space for the representation space.
Use the Invariants command to calculate the invariants of this representation.
See Also
DifferentialGeometry, Tensor, Tools, LieAlgebras, Invariants, GenerateForms, GenerateSymmetricTensors, Query, Representation
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